So - it appears to work now, and I’m not exactly sure why.
I think the final fix was specifying this in my gemfile:
gem ‘omniauth-twitch’, git: ‘git://github.com/WebTheoryLLC/omniauth-twitch.git’
Oddly, that didn’t seem to change anything. I then altered the route slightly and it worked perfectly. I was then able to alter the route BACK to what it was, and it still works.
I think I may not have restarted my server in between those steps, and I’m assuming that specifying the gem was the key here. (I was trying to be diligent about restarting the server whenever I made a change, but I can’t be positive I did right after that step).
Thanks for the help everyone!